Difference between revisions of "2024 SSMO Speed Round Problems/Problem 6"
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| + | Let <math>a_n</math> equal <math>1</math> if the nth house is occupied and <math>0</math> if the nth house is unoccupied. Thus, we are trying to find <math>\mathbb{E}[a_1+a_2+a_3+a_4] = 4\mathbb{E}[a_1]</math> by the Linearity of Expectation. Instead of calculating the probability that a single house is occupied, we instead calculate the probability that it is unoccupied. Each person has a <math>\frac{1}{4}</math> chance of residing in that house, and thus a <math>\frac{3}{4}</math> chance of not residing in that house. The probability that none of the people stay in that house is <math>\left (\frac{3}{4} \right)^4 = \frac{81}{256}</math>. Thus, the probability that any given house is occupied, or <math>\mathbb{E}[a_1]</math>, is <math>1 - \frac{81}{256} = \frac{175}{256}</math>. Thus, our answer is <math>4\cdot \frac{175}{256} = \frac{175}{64}</math>, <math>175+64=\boxed{239}</math>. | ||
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| + | -Vivdax | ||
Latest revision as of 19:41, 2 May 2025
Problem
There are 
people and houses. Each person independently randomly chooses a house to live in. The expected number of inhabited houses can be expressed as 
, where 
and 
are relatively prime positive integers. Find 
.
Solution
Let 
equal 
if the nth house is occupied and 
if the nth house is unoccupied. Thus, we are trying to find ![$\mathbb{E}[a_1+a_2+a_3+a_4] = 4\mathbb{E}[a_1]$](http://latex.artofproblemsolving.com/2/8/b/28b16f035372bc7b85c7c31849f99abdb0d0c0ee.png)
by the Linearity of Expectation. Instead of calculating the probability that a single house is occupied, we instead calculate the probability that it is unoccupied. Each person has a 
chance of residing in that house, and thus a 
chance of not residing in that house. The probability that none of the people stay in that house is 
. Thus, the probability that any given house is occupied, or ![$\mathbb{E}[a_1]$](http://latex.artofproblemsolving.com/4/b/d/4bd93b9d2ffbcd175bbb28d9a3b26778ff136e06.png)
, is 
. Thus, our answer is 
, 
.
-Vivdax
